The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm

The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is

Angle between y² = x and x² = y at the origin is

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\) 

The function sin⁴ x + cos⁴ x is increasing in the interval

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

Equation of the normal to the curve y = 2x²+3 sin x at x = 0 is __________

The least value of a when f f(x) = x² + ax + 1 is increasing on (1, 2) is __________

The angle made by any tangent to the curve y = x⁵ + 8x + 1 with the X-axis is a __________

The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________

The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________

The function -3x+12 is ________ function on R.

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

Find the angle of intersection of the curve y = sin x with the positive x -axis.

Find the slope of the tangent to the following curves at the respective given points
y = x⁴ + 2x² − x at x = 1

A truck travels on a toll road with a speed limit of 80 km/hr. The truck completes a 164 km journey in 2 hours. At the end of the toll road the trucker is issued with a speed violation ticket. Justify this using the Mean Value Theorem.

The volume of a cylinder is given by the formula V = πr² h. Find the greatest and least values of V if r + h = 6.

Evaluate the limit  \(\underset{x\rightarrow 0}{lim}(\frac{sin \ mx}{x})\)

A particle moves in a line so that x =\(\sqrt { t } \). Show that the acceleration is negative and proportional to the cube of the velocity.

Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.

Find the equation of the tangent to the curve y²=4x+5 and which is parallel to y=2x+7

Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

Find the absolute extreme of the function f(x) = x²-2x+2 on the closed interval [0, 3]

For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2

The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.

Prove using the Rolle’s theorem that between any two distinct real zeros of the polynomial \(a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\) there is a zero of the polynomial \(na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\)

Suppose that for a function f(x), f'(x) ≤ 1for all 1 ≤ x ≤ 4. Show that f(4) - f(1) ≤ 3.

Evaluate the limit \(\underset{x\rightarrow 0^{+}}{lim} (\frac{sin \ x}{x^{2}})\)

Evaluate: \(\underset{x\rightarrow 0^{+}}{lim}(\frac{1}{x}-\frac{1}{e^{x}-1})\).

If an initial amount A₀ of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is \(A={ A }_{ 0 }{ \left( 1+\frac { r }{ n } \right) }^{ nt }\). If the interest is compounded continuously, (that is as n ➝∞), show that the amount after t years is A = A_oe^rt.

Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].

The ends of a rod AB which is 5 m long moves along two grooves OX, OY which at the right angles. If A moves at a constant speed of \(\frac { 1 }{ 2 } \) m/sec, what is the speed of B, when it is 4m from O?

Verify Rolle’s theorem for f(x)=ex sinx,\(0\le x\le \pi \)

A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.

A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A 10 kilometres to the north of P and traveling at 80 km/hr, while car B is 15 kilometres to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing?

A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds

Find the equation of the tangent and normal to the Lissajous curve given by x = 2cos 3t and y = 3sin 2t, t ∈ R

Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x +12y = 12.

Find the intervals of monotonicities and hence find the local extremum for the following function:
f(x) = sin x cos x + 5, x ∈ (0,2π)

Prove that among all the rectangles of the given area square has the least perimeter.